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Describe some of the factors that shaped President Kennedy's foreign policy?

He inherited a Cold War, of course, which was then very close to the peak of its cold ferocity. That included more than one crisis over the status of Berlin and Eastern Europe generally. The amity at Potsdam had evaporated in the 50s, when Stalin declined to lift the boot of the Red Army from the neck of Eastern Europe, and the Americans had recruited West Germany -- Germans generally hated the Russians -- into being their front line in the defense against the USSR. The Berlin Airlift was recent memory (1949), and the Berlin Wall was built early in his term (1961). There had been no final resolution fo the status of occupied Germany and its occupied capital, Berlin, which had led to the founding of West and East Germany in hte late 1940s. Berlin continued to be an occupied city, with profound tensions between Soviet and American forces, which faced each other, mere feet apart:
http://www.army.mil/article/46993/Standoff_in_Berlin__October_1961/
There was profound anxiety over Soviet development of ICBMs -- in fact, Kennedy had exploited this anxiety -- the so-called "missile gap" -- to outflank Nixon to the right in the election of 1960, implying that Eisenhower and Nixon were old fuddies asleep at the switch with respect to Space Age military tech. Sputnik had been launched in 1957, and everyone knew if the Soviets could put a satellite in orbit from Kazakhstan, they could just as easily loft a nuclear bomb from the same place into New York City. Americans had barely gotten over the shock of knowing the Soviets actually had The Bomb, a bit of superduper supersecret (they thought) tech that only American ingenuity had been able, with great effort, to develop. The fact that the Soviets had duplicated the weapon, with the help of US and British traitors,
http://www.atomicarchive.com/Bios/Fuchs.shtml
...had led to a profound insecurity and suspicion of spies and subversives -- symbolized most clearly by the McCarthy hearings of the mid-1950s. Both the Soviet anger over American forward presence in West Germany and Turkey, and American anxiety over ICBMs, helped to precipitate the Cuban Missile Crisis of 1962:
http://www.youtube.com/watch?v=W50RNAbmy3M
An additional issue, for John Kennedy, was that he was young and inexperienced. He was the youngest man ever elected President, at 43, about ten years younger than is typical. He had been a mere lieutenant j.g. in the war, commanding a PT boat, and was succeeding General Eisenhower, who had been Supreme Commander in Europe during the war. So he (and the nation) felt he had big shoes to fill, particularly in foreign policy. That made him more aggressive than his experience warranted, and he made major mistakes -- the Vienna conference with Khruschev in 1961, the Bay of Pigs invasion, aspects of the Cuban Missile Crisis, and of course starting America's involvement in Vietnam. He didn't particularly get along with the Congressional establishment, which saw him as young and clumsy, not to mention too interested in chasing women and dressing smart. Partly to compensate, and because the press -- which was young and restless, like him -- adored him and actively promoted it as a brilliant move, he brought in to advise him a whole new crew, his "Best And Brightest," and fostered the notion that he represented a "New Frontier." (Promoting this image is one reason he proposed the Apollo Moon program:
http://www.youtube.com/watch?v=g25G1M4EXrQ
Finally, another difficult issue he faced is that he barely won election in 1960. (Indeed, arguments have been made that if the Daley Machine in Chicago had not "voted the graveyard" -- i.e. committed voter fraud -- he would not have won at all.) He was certainly popular as a young and dynamic man, but a great chunk of the public (and almost all the political establishment) saw him as a clueless whippersnapper who fronted for age-old corrupt Democratic machines -- for the Daley and Pendergast machines, old Joe Kennedy's moonshine money, for the deeply corrupt cities and labor unions. There was a deep sense of unease that he was not strong enough to be his own man, and not experienced enough to navigate the US through the murky waters of the Cold War.
One final point not appreciated at the time was that Kennedy inherited a lot of secrets from Eisenhower. One was the Venona transcripts, which showed just how extensive Soviet wartime spying had been in the US. These had proved the guilt of the Rosenbergs, for example, but could not be revealed in public without revealing to the Soviets this intelligence coup. Another was the Corona spy satellite program, which kept secret tabs on Soviet missile development. Still another was a series of extremely secret and highly provocative spy overflights of the USSR by the USAF, sometimes in nuclear bombers, later on with specialized spy planes, such as the U-2. (A crisis would be provoked when Francis Gary Powers, a U-2 pilot, was shot down by the Soviets over Soviet territory, and did not kill himself as he was supposed to.) This left Kennedy in the awkward position of knowing stuff about the USSR the sources of which he could not reveal. Eisenhower could get away with just asserting he knew things -- he was old enough, and had been part of wartime Soviet-American planning, and people just assumed he was canny enough to just know things. Kennedy couldn't do that -- but neither could he reveal the technical means that let him know things.
All of these factors influenced his foreign policy decisions, a few for the better, most for the worse. In the area of foreign policy, it seems difficult to argue that the United States wouldn've been better off had Nixon won the election of 1960.